Type: | Package |
Title: | Unbiased Central Moment Estimates |
Version: | 1.0.1 |
Maintainer: | Inna Gerlovina <innager@berkeley.edu> |
Description: | Calculates one-sample unbiased central moment estimates and two-sample pooled estimates up to 6th order, including estimates of powers and products of central moments. Provides the machinery for obtaining unbiased central moment estimators beyond 6th order by generating expressions for expectations of raw sample moments and their powers and products. Gerlovina and Hubbard (2019) <doi:10.1080/25742558.2019.1701917>. |
Depends: | R (≥ 3.4.0) |
Imports: | stats, utils |
License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
Encoding: | UTF-8 |
RoxygenNote: | 7.3.1 |
Suggests: | knitr, rmarkdown, testthat |
VignetteBuilder: | knitr |
NeedsCompilation: | no |
Packaged: | 2024-10-02 07:27:24 UTC; innars |
Author: | Inna Gerlovina [aut, cre], Alan Hubbard [aut] |
Repository: | CRAN |
Date/Publication: | 2024-10-02 08:10:02 UTC |
Generate symbolic expression for expectation
Description
Generate a string with symbolic expression for expectation of powers and products of non-central (raw) sample moments of an arbitrary order.
Usage
one_combination(powvect, smpsize = "n")
Arguments
powvect |
vector of non-negative integers representing exponents
|
smpsize |
symbol to be used for sample size. Defaults to |
Details
For a zero-mean random variable X
and a sample X_1, \dots,
X_n
, find E(\bar{X}^{j_1} \overline{X^2}^{j_2}
\overline{X^3}^{j_3} \cdots \overline{X^m}^{j_m})
, where overline{X^k} = 1/n
\sum_{i = 1}^n X_i^{k}
is a k
'th
non-central sample moment. The expression is given in terms of sample size
and true moments \mu_k
of X
. These expectations can
subsequently be used for generating unbiased central moment estimators of an
arbitrary order, Edgeworth expansions, and possibly solving other
higher-order problems.
Value
A string representing a symbolic expression for further processing using computer algebra (e.g. with Sage or SymPy), for calculating numeric values, or to be rendered with Latex.
Examples
one_combination(c(5, 0, 2, 1))
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products up to specified order.
Usage
uM(smp, order)
Arguments
smp |
sample. |
order |
highest order of the estimates to calclulate. Estimates of lower orders will be included. |
Details
Unbiased estimates up to the 6th order can be calculated. Second and third
orders contain estimates of the variance and third central moment, fourth
order includes estimates of fourth moment and squared variance
(\mu_2^2
), fifth order - of fifth moment and a product of
second and third moments (\mu_2 \mu_3
), sixth order - of
sixth moment, a product of second and fourth moments (\mu_2
\mu_4
), squared third moment (\mu_3^2
), and
cubed variance (\mu_2^3
).
Value
A named vector of estimates of central moments and their powers and
products up to order
. The highest order available is 6th. The names
of the elements are "M2", "M3", "M4", "M5", "M6"
for corresponding
central moments, "M2M3", "M2M4"
for products of the moments (second
and third, second and fourth), and "M2pow2", "M2pow3", "M3pow2"
for
powers of the moments - corresponding to estimates of squared variance,
cubed variance, and squared third moment.
References
Gerlovina, I. and Hubbard, A.E. (2019). Computer algebra and algorithms for unbiased moment estimation of arbitrary order. Cogent Mathematics & Statistics, 6(1).
See Also
uMpool
for two-sample pooled estimates.
Examples
smp <- rgamma(10, shape = 3)
uM(smp, 6)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM2(m2, n)
Arguments
m2 |
naive biased variance estimate |
n |
sample size. |
Value
Unbiased variance estimate.
See Also
Other unbiased estimates (one-sample):
uM2M3()
,
uM2M4()
,
uM2pow2()
,
uM2pow3()
,
uM3()
,
uM3pow2()
,
uM4()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
m <- c(m, mean((smp - m[1])^2))
uM2(m[2], n) - var(smp)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM2M3(m2, m3, m5, n)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m5 |
naive biased fifth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of a product of second and third central moments
\mu_2 \mu_3
, where \mu_2
and
\mu_3
are second and third central moments respectively.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M4()
,
uM2pow2()
,
uM2pow3()
,
uM3()
,
uM3pow2()
,
uM4()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:5) {
m <- c(m, mean((smp - m[1])^j))
}
uM2M3(m[2], m[3], m[5], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM2M3pool(m2, m3, m5, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m5 |
naive biased fifth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of a product of second and third central moments
\mu_2 \mu_3
, where \mu_2
and
\mu_3
are second and third central moments respectively.
See Also
Other pooled estimates (two-sample):
uM2M4pool()
,
uM2pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(5)
for (j in 2:5) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM2M3pool(m[2], m[3], m[5], nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM2M4(m2, m3, m4, m6, n)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of a product of second and fourth central moments
\mu_2 \mu_4
, where \mu_2
and
\mu_4
are second and fourth central moments respectively.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2pow2()
,
uM2pow3()
,
uM3()
,
uM3pow2()
,
uM4()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:6) {
m <- c(m, mean((smp - m[1])^j))
}
uM2M4(m[2], m[3], m[4], m[6], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM2M4pool(m2, m3, m4, m6, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of a product of second and fourth central moments
\mu_2 \mu_4
, where \mu_2
and
\mu_4
are second and fourth central moments respectively.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(6)
for (j in 2:6) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM2M4pool(m[2], m[3], m[4], m[6], nx, ny)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM2pool(m2, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled variance estimate.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
m2 <- mean(c((smpx - mean(smpx))^2, (smpy - mean(smpy))^2))
uM2pool(m2, nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM2pow2(m2, m4, n)
Arguments
m2 |
naive biased variance estimate |
m4 |
naive biased fourth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of squared variance \mu_2^2
, where
\mu_2
is a variance.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2M4()
,
uM2pow3()
,
uM3()
,
uM3pow2()
,
uM4()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:4) {
m <- c(m, mean((smp - m[1])^j))
}
uM2pow2(m[2], m[4], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM2pow2pool(m2, m4, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m4 |
naive biased fourth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of squared variance \mu_2^2
, where
\mu_2
is a variance.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(4)
for (j in 2:4) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM2pow2pool(m[2], m[4], nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM2pow3(m2, m3, m4, m6, n)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of cubed second central moment
\mu_2^3
, where \mu_2
is a variance.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2M4()
,
uM2pow2()
,
uM3()
,
uM3pow2()
,
uM4()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:6) {
m <- c(m, mean((smp - m[1])^j))
}
uM2pow3(m[2], m[3], m[4], m[6], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM2pow3pool(m2, m3, m4, m6, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of cubed variance central moment
\mu_2^3
, where \mu_2
is a variance.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pool()
,
uM2pow2pool()
,
uM3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(6)
for (j in 2:6) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM2pow3pool(m[2], m[3], m[4], m[6], nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM3(m3, n)
Arguments
m3 |
naive biased third central moment estimate |
n |
sample size. |
Value
Unbiased estimate of a third central moment.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2M4()
,
uM2pow2()
,
uM2pow3()
,
uM3pow2()
,
uM4()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:3) {
m <- c(m, mean((smp - m[1])^j))
}
uM3(m[3], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM3pool(m3, n_x, n_y)
Arguments
m3 |
naive biased third central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of a third central moment.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(3)
for (j in 2:3) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM3pool(m[3], nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM3pow2(m2, m3, m4, m6, n)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of squared third central moment
\mu_3^2
, where \mu_3
is a third central
moment.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2M4()
,
uM2pow2()
,
uM2pow3()
,
uM3()
,
uM4()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:6) {
m <- c(m, mean((smp - m[1])^j))
}
uM3pow2(m[2], m[3], m[4], m[6], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM3pow2pool(m2, m3, m4, m6, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of squared third central moment
\mu_3^2
, where \mu_3
is a third central
moment.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM4pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(6)
for (j in 2:6) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM3pow2pool(m[2], m[3], m[4], m[6], nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM4(m2, m4, n)
Arguments
m2 |
naive biased variance estimate |
m4 |
naive biased fourth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of a fourth central moment.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2M4()
,
uM2pow2()
,
uM2pow3()
,
uM3()
,
uM3pow2()
,
uM5()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:4) {
m <- c(m, mean((smp - m[1])^j))
}
uM4(m[2], m[4], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM4pool(m2, m4, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m4 |
naive biased fourth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of a fourth central moment.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM3pow2pool()
,
uM5pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(4)
for (j in 2:4) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM4pool(m[2], m[4], nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM5(m2, m3, m5, n)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m5 |
naive biased fifth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of a fifth central moment.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2M4()
,
uM2pow2()
,
uM2pow3()
,
uM3()
,
uM3pow2()
,
uM4()
,
uM6()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:5) {
m <- c(m, mean((smp - m[1])^j))
}
uM5(m[2], m[3], m[5], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM5pool(m2, m3, m5, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m5 |
naive biased fifth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Pooled estimate of a fifth central moment.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM6pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(5)
for (j in 2:5) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM5pool(m[2], m[3], m[5], nx, ny)
Unbiased central moment estimates
Description
Calculate unbiased estimates of central moments and their powers and products.
Usage
uM6(m2, m3, m4, m6, n)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n |
sample size. |
Value
Unbiased estimate of a sixth central moment.
See Also
Other unbiased estimates (one-sample):
uM2()
,
uM2M3()
,
uM2M4()
,
uM2pow2()
,
uM2pow3()
,
uM3()
,
uM3pow2()
,
uM4()
,
uM5()
Examples
n <- 10
smp <- rgamma(n, shape = 3)
m <- mean(smp)
for (j in 2:6) {
m <- c(m, mean((smp - m[1])^j))
}
uM6(m[2], m[3], m[4], m[6], n)
Pooled central moment estimates - two-sample
Description
Calculate pooled unbiased estimates of central moments and their powers and products.
Usage
uM6pool(m2, m3, m4, m6, n_x, n_y)
Arguments
m2 |
naive biased variance estimate |
m3 |
naive biased third central moment estimate |
m4 |
naive biased fourth central moment estimate |
m6 |
naive biased sixth central moment estimate |
n_x |
number of observations in the first group. |
n_y |
number of observations in the second group. |
Value
Unbiased estimate of a sixth central moment.
See Also
Other pooled estimates (two-sample):
uM2M3pool()
,
uM2M4pool()
,
uM2pool()
,
uM2pow2pool()
,
uM2pow3pool()
,
uM3pool()
,
uM3pow2pool()
,
uM4pool()
,
uM5pool()
Examples
nx <- 10
ny <- 8
shp <- 3
smpx <- rgamma(nx, shape = shp) - shp
smpy <- rgamma(ny, shape = shp)
mx <- mean(smpx)
my <- mean(smpy)
m <- numeric(6)
for (j in 2:6) {
m[j] <- mean(c((smpx - mx)^j, (smpy - my)^j))
}
uM6pool(m[2], m[3], m[4], m[6], nx, ny)
Pooled central moment estimates - two-sample
Description
Calculate unbiased pooled estimates of central moments and their powers and products up to specified order.
Usage
uMpool(smp, a, order)
Arguments
smp |
sample. |
a |
vector of the same length as |
order |
highest order of the estimates to calclulate. Estimates of lower orders will be included. |
Details
Pooled estimates up to the 6th order can be calculated. Second and third
orders contain estimates of the variance and third central moment, fourth
order includes estimates of fourth moment and squared variance
(\mu_2^2
), fifth order - of fifth moment and a product of
second and third moments (\mu_2 \mu_3
), sixth order - of
sixth moment, a product of second and fourth moments (\mu_2
\mu_4
), squared third moment (\mu_3^2
), and
cubed variance (\mu_2^3
).
Value
A named vector of estimates of central moments and their powers and
products up to order
. The highest order available is 6th. The names
of the elements are "M2", "M3", "M4", "M5", "M6"
for corresponding
central moments, "M2M3", "M2M4"
for products of the moments (second
and third, second and fourth), and "M2pow2", "M2pow3", "M3pow2"
for
powers of the moments - corresponding to estimates of squared variance,
cubed variance, and squared third moment.
References
Gerlovina, I. and Hubbard, A.E. (2019). Computer algebra and algorithms for unbiased moment estimation of arbitrary order. Cogent Mathematics & Statistics, 6(1).
See Also
uM
for one-sample unbiased estimates.
Examples
nsmp <- 23
smp <- rgamma(nsmp, shape = 3)
treatment <- sample(0:1, size = nsmp, replace = TRUE)
uMpool(smp, treatment, 6)